MODELLING AND SIMULATION OF STOCHASTIC SYSTEMS

GROUP PROJECT REPORT

DATE: 10/06/2014

DUE DATE: 13/06/2014

BY

Emmanuel Paul Malunga (a1656259)

Laveti Tejaswi (a1659176)

Florence Sitithana Ndenguma (a1656267)

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Table of Contents1.0 Aim 1

2.0 Objectives 1

3.0 Assumptions 1

4.0 Key Data – Literature Review 1

5.0 Brief background. 1

6.0 Methodology 2

7.0 Distributions used in the project 2

8.0 RESULTS AND DISCUSSION 2

9.0 Conclusion 5

10.0 Appendices 7

11.0 References 11

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List of Figures

1. Figure 1 – Population of Adelaide 32. Figure 2 – Vehicles Bar Graph 4 3. Figure 3 – Trees Bar Graph 44. Figure 4 - Carbon Dioxide Bar Graph 55. Figure 5 – Carbon Dioxide Plot 6 6. Figure 6 - Box plot 20 years 77. Figure 7 - Box plot 40 years 78. Figure 8 - Carbon dioxide levels: Histogram plot 20 years 89. Figure 9 - Carbon dioxide levels: Histogram plot 40 years 810. Figure 10 - Population Plot 911. Figure 11 - Trees Plot 912. Figure12 - Vehicles Plot 1013. Figure 13 – Confidence Intervals for 40 years 10

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1.0 AimThe aim of this project is to determine the levels of Carbon-dioxide (CO2) in the city of Adelaide for the coming 4-5 decades.

2.0 Objectives

1. To estimate the various factors this may have a detrimental effect in the levels of CO2 in Adelaide.

2. To determine the various factors this may increase the levels of CO2 in Adelaide.

3.0 AssumptionsThe following assumptions are applicable in this project:

1. The family size for Adelaide is 4.2. Number of 2 vehicles per family is 1.3. The number of trees to be planted every year is 100,000.

4.0 Key Data – Literature Review1. Average adult person produces 0.74m3 of carbon dioxide per hour.2. The population of Adelaide will grow by 8% per decade for this period.3. Initial population for Adelaide is 13 million.4. All readings I the project considered are for an average adult.

5.0 Brief background.Any living thing inhales oxygen and exhales carbon dioxide. Some activities that increase the production of carbon dioxide include vehicles, industrial activities such as power generation plants, deforestation. On the other hand, there are other activities that help to reduce the amount of carbon dioxide in the atmosphere. These include trees (these are the only natural purifiers that take in carbon dioxide and produce oxygen), use automobiles which run on Hydrogen, Nitrogen. Other new technologies also reduce the production of carbon dioxide. Such technologies include solar powered vehicles and power generation from wind.

Increasing population has resulted in the increase in demand for mobility and power energy. People normally move from one place to another by using vehicles. The increasing population therefore results in the increased number of vehicles used. This may have an impact on the amount of carbon dioxide being produced. On the other hand, trees which are a natural purifier of the atmosphere are on the decrease as more and more development activities are taking place as the increased population is trying to meet the day to day requirements.

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It is for these reasons that this project was conducted with the aim of estimating the carbon dioxide levels in Adelaide in the next four to five decades. The project also studies the effect of increasing plantations by 100,000 trees every year on the levels of carbon dioxide.

6.0 Methodology Due to time limitations, data used in this project was based on research from the internet and assumptions listed above. The data was computed using MATLAB computer package.

7.0 Distributions used in the projectThere were mainly two types of distributions that were used in this project namely; binomial distribution and Poisson distribution.

Binomial distribution was used when calculating the population increase because the initial population is known and the associated factors are also known. These associated factors are birth rate, death rate and the population increase. Poisson distribution was used when calculating the increase in the number of trees as well as the increase in the number of vehicles because the actual rate of increase is not known. Much as it was assumed that 100,000 trees will be planted every year, it is not certain that all the trees planted will survive. As such the actual number of trees that will survive is not known hence the use of Poisson distribution. Similarly, the actual increase in number of vehicles is not known.

8.0 RESULTS AND DISCUSSIONSeveral runs were made on the program using MATLAB.

Results were presented as histograms and plots as shown in the following pages.

3.1 Population in Adelaide

Results show that the population of Adelaide is increasing exponentially as shown in the histogram plotted (Figure 1) that follows.

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Figure 1 – Population of Adelaide

Similarly, the plot also shows a similar trend as shown in figure … in the appendix.

This agrees with the assumptions indicated in section 3.0 where increase in population was estimated at 8% per annum. It is expected that in the coming 4-5 decades the population of Adelaide will be approximately 2 million. This follows that there will be more carbon dioxide being emitted since it is assumed that on average an average adult produces 0.74 m3 of carbon dioxide per hour. In simple mathematics it follows that approximately (2,000,000 x 0.74) = 1,480,000 m3 of carbon dioxide will be emitted per hour in the next 4-5 decades.

3.2 Vehicles in Adelaide

A histogram that was plotted during the simulation runs show that the number of vehicles also keeps on increasing as shown on the histogram and plot respectively. The histogram is shown as figure 2 below.

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Figure 2 – Vehicles Bar Graph

3.3 Trees

Simulation results show that trees will be increasing as shown in graph ……

Figure 3 – Trees Bar Graph

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It can be noted from the carbon dioxide bar graph that as the number of trees increases, the level of carbon dioxide decreases. This is evident in the carbon dioxide bar graph shown…. But at the same time even though trees are increasing, resulting in a decrease in carbon dioxide levels, the increase in the number of vehicles results in the increase in the amount of carbon dioxide being emitted. It can therefore be noted that the levels of carbon dioxide keep on fluctuating due to increase in the population, number of vehicles as well as number of trees. A plot of carbon dioxide against time shown in figure …. signifies these observations. It can however be noted that the increase in number of trees has a very big impact in the carbon dioxide levels. Before the increase in plantations, high levels of carbon dioxide are noted. As more and more trees are being planted over the years, the carbon dioxide levels reduce tremendously.

Figure 4 - Carbon Dioxide Bar Graph

9.0 ConclusionIt can be concluded from the simulation results that levels of carbon dioxide are affected by the increase in population, the increase in number of vehicles and the increase in number of trees planted. Increase in population and increase in number of vehicles results in increase in carbon dioxide levels. The increase in number of trees results in the decrease in carbon dioxide levels. The plot that follows shows the fluctuating carbon dioxide levels over the years for the period under study (40 years).

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Figure 5 – Carbon Dioxide Plot

It can therefore be concluded that there are basically three factors that affect the levels of carbon dioxide in Adelaide. The increase in population and number of vehicles results in the increase in carbon dioxide levels whereas the increase in the number of trees planted results in the decrease in the levels of carbon dioxide. The plot shows that on average, the carbon level is between 6 to 8 tons.

In order to keep the carbon dioxide levels to a minimum, the following suggestions are made:

1. Increase the number of trees to be planted2. Reduce the number of vehicles to be used by introducing vehicles that use

other forms of energy apart from petrol and diesel

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10.0 AppendicesThis section shows some of a number of graphs that were obtained during the various simulation runs.

Figure 6 - Box plot 20 years

Figure 7 - Box plot 40 years

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Figure 8 - Carbon dioxide levels: Histogram plot 20 years

Figure 9 - Carbon dioxide levels: Histogram plot 40 years

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Figure 10 - Population Plot

Figure 11 - Trees Plot

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Figure12 - Vehicles Plot

Figure 13 – Confidence Intervals for 40 years

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11.0 References

http://www.abs.gov.au/ausstats/abs@.nsf/mf/9309.0 ref 8

Number of co2 with respect to km Ref 7

http://textilewastediversion.com/blog/

Number of Vehicles Reference

http://www.abs.gov.au/ausstats/abs@.nsf/mf/9208.0/ Ref 6

Vehicle reference:

http://www.carbonfootprint.com/calculator.aspx Ref 5

Population reference: https://www.infrastructure.gov.au/infrastructure/pab/soac/files/factsheets_2013/Adelaide_Factsheet_FA.pdf Ref 4

Tree purifying oxygen:

http://www.coloradotrees.org/benefits.htm#10 Ref 3

Calculation of co2:

p. 44, Kapit, W., et al., 1987: The Physiology Coloring Book. HarperCollins. 154 pp. Ref 2

Calculation of co2:

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p. 151, Berkow,, R., et al., 1997: The Merck Manuel for Medical Information: Home Edition. Merck & Co, publishers, 1509 pp. Ref 1

Main reference:

www.EngineeringToolBox.com

Appendix: Simulation Code

clear()kmsNorth = 40.0kmsEast = 10.0treesPHectare = 100;propForested = 0.25;hPkm = 100;nSims= 1000;areaUnderForest = propForested*kmsNorth*kmsEast;carsPerPerson = 1.0 propDeath = 0.01;propBirth = 0.02;popIncrease = 0.08;propvehicles = 0.2;proptrees= 0.1;plantationProb = 0.20;plantationSize =100000; maxYears = 40amountco2emittedperson= 0.9*365amountco2emittedVehicle = 0.9amountco2absorbedtrees = 0.6popAdelaide = zeros(1,maxYears);levelofco2 = zeros(maxYears,nSims);ntrees= zeros(1,maxYears);nVehicles= zeros(1,maxYears);co2absorbedtrees= zeros(1,maxYears);co2Emittedvehicle=zeros(1,maxYears);co2Emittedpeople=zeros(1,maxYears); for sim =1:nSims popAdelaide(1) = 1370000;levelofco2(1,sim) = 1000000;ntrees(1) = areaUnderForest*hPkm*treesPHectarenVehicles(1) = popAdelaide(1)*carsPerPerson;co2Emittedpeople= amountco2emittedperson*popAdelaide(1)co2Emittedvehicle = nVehicles(1) * amountco2emittedVehicleco2absorbedtrees =amountco2absorbedtrees* ntrees(1); for nYear = 2: maxYears nDeaths = binornd(popAdelaide(nYear-1), propDeath);

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nBirths = binornd(popAdelaide(nYear-1), propBirth); nPlantations = poissrnd(plantationProb); nVehiclesincrease = poissrnd(propvehicles); ntrees(nYear) = ntrees(nYear-1)+ nPlantations*plantationSize; popAdelaide(nYear) = popAdelaide(nYear-1) -nDeaths + nBirths; nVehicles(nYear) = popAdelaide(nYear)*carsPerPerson+(nVehiclesincrease *nVehicles(nYear-1)); levelofco2(nYear,sim)= popAdelaide(nYear)+nVehicles(nYear)-ntrees(nYear); endend h = figure; plot(levelofco2); title(' plot of levels of Carbon-Dioxide over Number of Years'); xlabel('number of years'); ylabel(' levels of co2( tons)'); saveas(h,'CO2','pdf'); mean1 = mean(levelofco2(:,:),2); stdev1 = std(levelofco2(:,:),0, 2); level = stdev1/sqrt(nSims)*1.96 ci = mean1+level;

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